?

Average Error: 0.1 → 0.2
Time: 12.6s
Precision: binary32
Cost: 13408

?

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\frac{\frac{0.125}{\pi}}{s} \cdot \left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{\frac{-r}{s}}}{r}\right) \]
(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))
↓
(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 PI) s)
  (+
   (exp (- (* -0.3333333333333333 (/ r s)) (log r)))
   (/ (exp (/ (- r) s)) r))))
float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}
↓
float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * (expf(((-0.3333333333333333f * (r / s)) - logf(r))) + (expf((-r / s)) / r));
}

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end
↓
function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / s) * Float32(exp(Float32(Float32(Float32(-0.3333333333333333) * Float32(r / s)) - log(r))) + Float32(exp(Float32(Float32(-r) / s)) / r)))
end

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end
↓
function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * (exp(((single(-0.3333333333333333) * (r / s)) - log(r))) + (exp((-r / s)) / r));
end

\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}
↓
\frac{\frac{0.125}{\pi}}{s} \cdot \left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{\frac{-r}{s}}}{r}\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.1

    \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right)} \]
    Proof

    [Start]0.1

    \[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    times-frac [=>]0.1

    \[ \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    times-frac [=>]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} \]

    associate-*l* [=>]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

    associate-/r* [=>]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

    metadata-eval [=>]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\color{blue}{0.125}}{\pi \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

    metadata-eval [<=]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\color{blue}{\frac{0.25}{2}}}{\pi \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

    associate-/r* [<=]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{0.25}{2 \cdot \left(\pi \cdot s\right)}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

    associate-*l* [<=]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.25}{\color{blue}{\left(2 \cdot \pi\right) \cdot s}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

    distribute-lft-out [=>]0.1

    \[ \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{3 \cdot s}}}{r}\right)} \]

    +-commutative [<=]0.1

    \[ \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \color{blue}{\left(\frac{e^{\frac{-r}{3 \cdot s}}}{r} + \frac{e^{\frac{-r}{s}}}{r}\right)} \]

  3. Applied egg-rr0.5

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125}{s \cdot \pi}\right)} - 1\right)} \cdot \left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

  4. Simplified0.2

    \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
    Proof

    [Start]0.5

    \[ \left(e^{\mathsf{log1p}\left(\frac{0.125}{s \cdot \pi}\right)} - 1\right) \cdot \left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

    expm1-def [=>]0.2

    \[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{s \cdot \pi}\right)\right)} \cdot \left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

    expm1-log1p [=>]0.2

    \[ \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

    associate-/r* [=>]0.2

    \[ \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

  5. Applied egg-rr0.2

    \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \left(\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r}} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

  6. Taylor expanded in s around 0 0.2

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

  7. Simplified0.2

    \[\leadsto \color{blue}{\frac{\frac{0.125}{\pi}}{s}} \cdot \left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
    Proof

    [Start]0.2

    \[ \frac{0.125}{s \cdot \pi} \cdot \left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

    associate-/l/ [<=]0.2

    \[ \color{blue}{\frac{\frac{0.125}{\pi}}{s}} \cdot \left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

  8. Final simplification0.2

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{\frac{-r}{s}}}{r}\right) \]

Alternatives

Alternative 1
Error0.2
Cost13408
\[\left(e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r} + \frac{e^{\frac{-r}{s}}}{r}\right) \cdot \frac{\frac{0.125}{s}}{\pi} \]
Alternative 2
Error0.2
Cost10208
\[\frac{0.125}{\pi \cdot s} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}\right) \]
Alternative 3
Error0.2
Cost10144
\[0.125 \cdot \frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \left(\pi \cdot r\right)} \]
Alternative 4
Error0.2
Cost10144
\[0.125 \cdot \frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot \left(s \cdot r\right)} \]
Alternative 5
Error17.8
Cost9792
\[\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)} \]
Alternative 6
Error29.1
Cost6880
\[\frac{\frac{0.125}{s}}{\pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{1}{r}\right) \]
Alternative 7
Error29.2
Cost3456
\[\frac{1}{s} \cdot \frac{\frac{0.25}{\pi}}{r} \]
Alternative 8
Error29.2
Cost3392
\[\frac{0.25}{r \cdot \left(\pi \cdot s\right)} \]
Alternative 9
Error29.2
Cost3392
\[\frac{0.25}{s \cdot \left(\pi \cdot r\right)} \]
Alternative 10
Error29.2
Cost3392
\[\frac{0.25}{\pi \cdot \left(s \cdot r\right)} \]
Alternative 11
Error29.2
Cost3392
\[\frac{\frac{0.25}{s}}{\pi \cdot r} \]
Alternative 12
Error29.2
Cost3392
\[\frac{\frac{\frac{0.25}{\pi}}{r}}{s} \]

Error

Reproduce?

herbie shell --seed 2023060 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))